We briefly summarize the weak hypocoercivity framework as well as the main result from [1]}, which along with [2]} is recommended for more details.
Let \((H,(\cdot ,\cdot ),\Vert \cdot \Vert )\)  be a separable Hilbert space with orthogonal decomposition \(H=H_1\oplus H_2\)  given by the orthogonal projection \(P:H\rightarrow H_1\) . Let \((L,D(L))\)  be a densely defined linear operator generating a \(C_0\) -semigroup \((T_t)_{t\ge 0}\)  on \(H\)  and let \(D\)  be a core of \((L,D(L))\) . Further assume that \(L\)  decomposes into \(L=S-A\)  on \(D\) , where \(S\)  is symmetric and \(A\)  is antisymmetric. This implies that \((S,D)\)  and \((A,D)\)  are closable; we denote their closures by \((S,D(S))\)  and \((A,D(A))\) , respectively. We assume the following:
